A Bayesian Non-parametric Approach to Survival Analysis Using Polya Trees

This paper presents a Bayesian non-parametric approach to survival analysis based on arbitrarily right censored data. The analysis is based on posterior predictive probabilities using a Polya tree prior distribution on the space of probability measures on [0, ∞). In particular we show that the estimate generalizes the classical Kaplanndash;Meier non-parametric estimator, which is obtained in the limiting case as the weight of prior information tends to zero.     

Non-parametric Bayesian analysis of clustered survival data

Clustered survival data are often modelled with frailty models which incorporate frailties to model the cluster specific heterogeneity and the dependence between observations in the same cluster. For the analysis of the frailty models, we propose Bayesian modelling with beta process prior on the cumulative hazard function and describe the details of the posterior computation. We demonstrate the method with two data sets using three different frailty distributions: gamma, log-normal and log-logistic distributions. We also empirically demonstrate the difficulty in checking the assumed frailty distribution with the posterior sample of the frailties.     

Non parametric Bayesian analysis of the two-sample problem with censoring

Testing for differences between two groups is a fundamental problem in statistics, and due to developments in Bayesian non parametrics and semiparametrics there has been renewed interest in approaches to this problem. Here we describe a new approach to developing such tests and introduce a class of such tests that take advantage of developments in Bayesian non parametric computing. This class of tests uses the connection between the Dirichlet process (DP) prior and the Wilcoxon rank sum test but extends this idea to the DP mixture prior. Here tests are developed that have appropriate frequentist sampling procedures for large samples but have the potential to outperform the usual frequentist tests. Extensions to interval and right censoring are considered and an application to a high-dimensional data set obtained from an RNA-Seq investigation demonstrates the practical utility of the method.     

Comparison of Regression Curves with Censored Responses

Abstract.  In this article, we introduce a procedure to test the equality of regression functions when the response variables are censored. The test is based on a comparison of Kaplan–Meier estimators of the distribution of the censored residuals. Kolmogorov–Smirnov- and Cramér–von Mises-type statistics are considered. Some asymptotic results are proved: weak convergence of the process of interest, convergence of the test statistics and behaviour of the process under local alternatives. We also describe a bootstrap procedure in order to approximate the critical values of the test. A simulation study and an application to a real data set conclude the paper.     

Bayesian nonparametric survival analysis using mixture of Burr XII distributions

ABSTRACT

Recently, the Bayesian nonparametric approaches in survival studies attract much more attentions. Because of multimodality in survival data, the mixture models are very common. We introduce a Bayesian nonparametric mixture model with Burr distribution (Burr type XII) as the kernel. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time datasets, we show the preference of this model and compare it with Dirichlet process mixture models with different kernels. The Markov chain Monte Carlo (MCMC) simulation methods to calculate the posterior distribution are used.     

Most powerful tests for use in matched pair experiments when data may be censored

Four distribution-free tests are developed for use in matched pair experiments when data may be censored: a bootstrap based on estimates of the median difference, and three rerandomization tests. The latter include a globally almost most powerful (GAMP) test which uses the original data and two modified Gilbert-Gehan tests which use the ranks. Computation time is reduced by using a binary count to generate subsamples and by restricting subsampling to the uncensored pairs. In Monte Carlo simulations against normal alternatives, mixed normal alternatives, and exponential alternatives, the GAMP test is most powerful with light censoring, the rank test is most powerful with heavy censoring. The bootstrap degenerates to the sign test and is least powerful.     

Bayes estimators of the reliability function and parameter of inverted exponential distribution using informative and non-informative priors

In this paper, we propose Bayes estimators of the parameter and reliability function of inverted exponential distribution under the general entropy loss function for complete, type I and type II censored samples. The proposed estimators have been compared with the corresponding maximum-likelihood estimators for their simulated risks (average loss over sample space).     

Bayesian fitting of a logistic dose-response curve with numerically derived priors

In this report we describe the Bayesian analysis of a logistic dose-response curve in a Phase I study, and we present two simple and intuitive numerical approaches to construction of prior probability distributions for the model parameters. We combine these priors with the expert prior opinion and compare the results of the analyses with those obtained from the use of alternative prior formulations.     

Caries on Permanent Teeth: A Non-parametric Bayesian Analysis

Most earlier epidemiological investigations of dental caries have been based on cross-sectional data. Subject-specific information of dental caries in the past, and the duration of exposure of each tooth to the oral environment, are obviously important factors also influencing the presence of dental caries in the future. This has led us to consider multivariate survival models in which the information about the tooth eruption and failure times are combined to assess caries risk. A non-parametric Bayesian intensity model is presented, reflecting, on the one hand, the within subject and between subject sources of variability, and a corresponding split of variability when considering the 28 permanent teeth. We analyse a data set consisting of the dental history of 240 boys, where the observations are based on predetermined dental examinations taking place approximately once every year. Markov chain Monte Carlo integration techniques are applied in the numerical work.     

Non-parametric Bayesian inference through MCMC method for Y-linked two-sex branching processes with blind choice

ABSTRACT

Dirichlet-process-based non-parametric Bayesian inference is developed for a Y-linked two-sex branching process with blind choice. This stochastic model is suitable for analysing the evolution of the number of carriers of two alleles of a Y-linked gene in a two-sex monogamous population where each female chooses her partner from among the male population without caring about his type (i.e. the allele he carries). The only data assumed to be available are the total number of females and males (regardless of their types) up to some generation and the numbers of each type of male in the last generation. A simulation method which is based on a Dirichlet process and a Gibbs sampler is developed to estimate the posterior distributions of the model's main parameters. Finally, the computational efficiency of the algorithm is illustrated with example simulations and an application to real data.     

A Class of Permutation Tests for the Equality of Two Marginal Survival Functions Using Paired Censored Data

We studied several test statistics for testing the equality of marginal survival functions of paired censored data. The null distribution of the test statistics was approximated by permutation. These tests do not require explicit modeling or estimation of the within-pair correlation, accommodate both paired data and singletons, and the computation is straightforward with most statistical software. Numerical studies showed that these tests have competitive size and power performance. One test statistic has higher power than previously published test statistics when the two survival functions under comparison cross. We illustrate use of these tests in a propensity score matched dataset.     

A Full Bayesian Non-parametric Analysis Involving a Neutral to the Right Process

Implementation of a full Bayesian non-parametric analysis involving neutral to the right processes (apart from the special case of the Dirichlet process) has been difficult for two reasons: first, the posterior distributions are complex and therefore only Bayes estimates (posterior expectations) have previously been presented; secondly, it is difficult to obtain an interpretation for the parameters of a neutral to the right process. In this paper we extend Ferguson & Phadia (1979) by presenting a general method for specifying the prior mean and variance of a neutral to the right process, providing the interpretation of the parameters. Additionally, we provide the basis for a full Bayesian analysis, via simulation, from the posterior process using a hybrid of new algorithms that is applicable to a large class of neutral to the right processes (Ferguson & Phadia only provide posterior means). The ideas are exemplified through illustrative analyses.     

A Bayesian model for multinomial sampling with misclassified data

In this paper the issue of making inferences with misclassified data from a noisy multinomial process is addressed. A Bayesian model for making inferences about the proportions and the noise parameters is developed. The problem is reformulated in a more tractable form by introducing auxiliary or latent random vectors. This allows for an easy-to-implement Gibbs sampling-based algorithm to generate samples from the distributions of interest. An illustrative example related to elections is also presented.     

Non-parametric Bayesian Estimation of a Spatial Poisson Intensity

A method introduced by Arjas & Gasbarra (1994) and later modified by Arjas & Heikkinen (1997) for the non-parametric Bayesian estimation of an intensity on the real line is generalized to cover spatial processes. The method is based on a model approximation where the approximating intensities have the structure of a piecewise constant function. Random step functions on the plane are generated using Voronoi tessellations of random point patterns. Smoothing between nearby intensity values is applied by means of a Markov random field prior in the spirit of Bayesian image analysis. The performance of the method is illustrated in examples with both real and simulated data.     

A semiparametric Bayesian model for repeatedly repeated binary outcomes

Summary.  We discuss the analysis of data from single-nucleotide polymorphism arrays comparing tumour and normal tissues. The data consist of sequences of indicators for loss of heterozygosity (LOH) and involve three nested levels of repetition: chromosomes for a given patient, regions within chromosomes and single-nucleotide polymorphisms nested within regions. We propose to analyse these data by using a semiparametric model for multilevel repeated binary data. At the top level of the hierarchy we assume a sampling model for the observed binary LOH sequences that arises from a partial exchangeability argument. This implies a mixture of Markov chains model. The mixture is defined with respect to the Markov transition probabilities. We assume a non-parametric prior for the random-mixing measure. The resulting model takes the form of a semiparametric random-effects model with the matrix of transition probabilities being the random effects. The model includes appropriate dependence assumptions for the two remaining levels of the hierarchy, i.e. for regions within chromosomes and for chromosomes within patient. We use the model to identify regions of increased LOH in a data set coming from a study of treatment-related leukaemia in children with an initial cancer diagnostic. The model successfully identifies the desired regions and performs well compared with other available alternatives.     

A Bayesian nonparametric estimator of a multivariate survival function

Using reinforced processes related to beta-Stacy process and generalized Pólya urn scheme jointly with a structure assumption about dependence, a Bayesian nonparametric prior and a predictive estimator for a multivariate survival function are provided. This estimator can be computed through an easy implementation of a Gibbs sampler algorithm. Moreover consistency of the estimator is studied.     

Relative efficiency of a censored experiment in terms of fisher information matrix

Fisher information matrix is used to quantify information loss in the randomly right-censored model, A real value approach alternative to the matrix approach of Turrero (1988) is presented for obtaining real valued measures of the relative efficiency of the censored experiment. Properties of the proposed measures are examined. The connection between both approaches and the Bayesian approach to this problem is also studied. Results in the paper are exemplified by considering grouped survival data.     

A regression method for censored inverse-Gaussian data

Multiple regression methods are considered for progressively right-censored inverse-Gaussian data. Maximum-likelihood estimators are derived using the EM algorithm, and their asymptotic distributional properties are presented. The methodology is demonstrated using two case illustrations, one of which involves the defective feature of the inverse-Gaussian distribution. The relationship of the methodology to regression methods for complete inverse-Gaussian samples and to other regression methods for censored survival data is discussed.     

Non-parametric Bayesian Inference for Integrals with respect to an Unknown Finite Measure

Abstract.  We consider the problem of estimating a collection of integrals with respect to an unknown finite measure μ from noisy observations of some of the integrals. A new method to carry out Bayesian inference for the integrals is proposed. We use a Dirichlet or Gamma process as a prior for μ , and construct an approximation to the posterior distribution of the integrals using the sampling importance resampling algorithm and samples from a new multidimensional version of a Markov chain by Feigin and Tweedie. We prove that the Markov chain is positive Harris recurrent, and that the approximating distribution converges weakly to the posterior as the sample size increases, under a mild integrability condition. Applications to polymer chemistry and mathematical finance are given.     

Extending the family of Bayesian bootstraps and exchangeable urn schemes

A Bayesian bootstrap for a finite population with censored observations is introduced. It is shown to reduce to the finite population Bayesian bootstrap if there is no censoring and to reduce to the censored data Bayesian bootstrap for a large population. A class of general urn schemes for simulating exchangeable sequences of variables is introduced which is connected to the bootstrap method.